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Results tagged with set-theory
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user 126667
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
10
votes
Set theory for category theory beginners
Personally I found the language of sets and classes confusing, just as you describe. I've never been sure precisely what operations on classes are allowed. For instance some textbooks mention the ca …
15
votes
2
answers
3k
views
How should we define "locally small"?
Let U be a Grothendieck universe, and U+ its successor universe (assume Grothendieck's universe axiom).
Everybody agrees that a U-small category is a category whose sets of objects and morphisms are …
- 25.2k
11
votes
Find a "natural" group that contains the quotient of the infinite symmetric group by the alt...
If one considers the distinguishing feature of the sign homomorphism $S_n \to \mathbb{Z}/2$ to be that it is the canonical map from $S_n$ to its abelianization, then there is nothing analogous for $S_ …
- 25.2k
24
votes
Accepted
Can we disallow finite choice?
You might want topos theory. A topos is something like the category of sets, but the internal logic of a general topos is much weaker than ZF; it need not even be Boolean. An example of a topos is t …
- 25.2k
2
votes
Model category structure on Set without axiom of choice
I came across the nlab page for the axiom COSHEP (category of sets has enough projectives) which seems to be just what's needed to obtain a model category structure, as usually understood, on Set with …
- 25.2k
8
votes
Accepted
Locally presentable categories, universes, and Vopenka's principle
In the Grothendieck universe approach to category theory, as you say, we replace all small sets with $\mathcal{U}$-small sets. Let's look at the definition of a locally presentable $\mathcal{U}$-categ …
- 25.2k
16
votes
3
answers
3k
views
Model category structure on Set without axiom of choice
There is a model category structure on Set in which the cofibrations are the monomorphisms, the fibrations are maps which are either epimorphisms or have empty domain, and the weak equivalences are th …
- 25.2k